Given that the perimeters of triangles A and B are equal, we need to set up an equation based on the perimeters.
Perimeter of Triangle A: \[ P_A = AB + BC + AC = (4x + 4) + (4x) + (4x) = 12x + 4 \]
Perimeter of Triangle B: \[ P_B = PQ + QR + PR = (4x + 3) + (5x + 2) + (4x) = (4x + 5x + 4x) + (3 + 2) = 13x + 5 \]
Since the perimeters are equal, we set \(P_A\) equal to \(P_B\): \[ 12x + 4 = 13x + 5 \]
Now, we solve for \(x\): \[ 12x + 4 - 4 = 13x + 5 - 4 \] \[ 12x = 13x + 1 \] \[ 12x - 13x = 1 \] \[ -x = 1 \] \[ x = -1 \]
Now that we have the value of \(x\), we can substitute it back to find the side lengths:
For Triangle A:
- \( AB = 4x + 4 = 4(-1) + 4 = -4 + 4 = 0 \)
- \( BC = 4x = 4(-1) = -4 \)
- \( AC = 4x = 4(-1) = -4 \)
For Triangle B:
- \( PQ = 4x + 3 = 4(-1) + 3 = -4 + 3 = -1 \)
- \( QR = 5x + 2 = 5(-1) + 2 = -5 + 2 = -3 \)
- \( PR = 4x = 4(-1) = -4 \)
The side lengths for both triangles yield invalid (non-positive) results, indicating no valid triangle can be formed with these side lengths when \(x = -1\).
It seems there was an inconsistency in the problem. If the side lengths must be positive, then the value for \(x\) should be re-evaluated in the context of the side length equations or other specifics might need revision. The given lengths cannot form triangles with the conditions that are stated or \(x\) should be limited to positive integers.
If you're looking for valid side lengths based on a different value of \(x\), please adjust the conditions or check if there's additional information I need.